The mass in the Coma cluster is approximately 10^15 M⊙ within a radius of 2 Mpc. Estimate the ra

step 1 We are going to use the ideal gas law pv equals n r t equals number of molecules times k times t

The mass in the Coma cluster is approximately 10^15 M⊙...

The mass in the Coma cluster is approximately $10^{15} M_{\odot}$ within a radius of $2 \mathrm{Mpc}$. Estimate the radius of a comoving volume of matter at the mean density which contains the same mass as the cluster. What is the temperature of the gas in the cluster, assuming it is isothermal? What would be the corresponding density profile of the cluster mass? The $x$-ray flux from bremsstrahlung is $5 \times 10^{37} \mathrm{~W}$ in the waveband $2-10 \mathrm{keV}$. What is the mass of hot gas in the cluster? Estimate roughly how long the gas would take to cool.

Key Concepts

Cosmic Mean Density and Comoving Volume

This concept involves understanding how to relate the mass contained within a specified region of the Universe to a comoving volume defined at the cosmic mean density. Comoving volume is a volume element that expands with the Universe, keeping the same set of matter particles. In problems like these, one estimates the radius of a sphere that, at the current cosmic average matter density, contains a specific mass—providing insight into how overdense a structure like a galaxy cluster is relative to the background density.

Virial Temperature and the Virial Theorem

The virial theorem relates the gravitational potential energy and kinetic energy in a bound system such as a galaxy cluster. Under the assumption of equilibrium, this theorem is used to estimate the temperature of the intracluster gas. The gas is heated by the gravitational potential well of the cluster, and its isothermal temperature reflects the balance between gravitational collapse and thermal pressure support, typically resulting in multi–keV temperatures.

Isothermal Assumption and Density Profile

Assuming that the gas in a cluster is isothermal simplifies the analysis by treating the temperature as constant throughout the cluster. This assumption helps determine the cluster’s density profile, which describes how the mass density decreases with radius. Often, models such as the King model or the more general beta?model are used to fit observed X-ray surface brightness profiles, implying a specific radial dependence of the density profile under hydrostatic equilibrium conditions.

Bremsstrahlung Emission in Hot Gas

Bremsstrahlung, or free–free emission, is a key mechanism for X–ray production in hot, ionized gas in clusters. The process involves electrons being decelerated in the electric fields of ions, producing a continuous spectrum of X–rays. The measured X–ray flux in a certain energy band can be used to infer properties of the emitting gas, such as density and total mass, by relating the emissivity of the gas to its temperature and density.

Cooling Time of the Intracluster Gas

The cooling time is the estimate of how long it would take the hot gas in the cluster to radiate away its thermal energy via processes like bremsstrahlung. It depends on the gas density, temperature, and the radiative cooling rate. In clusters, the cooling time is typically long compared to the age of the Universe, which explains why the hot gas in clusters remains at high temperatures; however, in the densest central regions, cooling can become significant and can lead to cooling flows.

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